非线性二阶常微分系统边值问题正解存在的充分必要条件 Alternative Title Necessary and Sufficient Condition for the Existence of positive solutions of Nonlinear Second-Order Ordinary Differential Systems 唐录天 Thesis Advisor 孙红蕊 2008-05-23 Degree Grantor 兰州大学 Place of Conferral 兰州 Degree Name 硕士 Keyword 非线性常微分系统 边值问题 最大值原理 正解 充分必要条件 上、下解 单调迭代 存在性 唯一性 Abstract 本文主要借助于非线性泛函分析中的上、下解方法，Schauder 不动点定理并结合单调迭代技巧，分别对两类非线性常微分系统正解存在的充分必要条件进行了研究。 第一章叙述了问题产生的背景和当前的研究进展。 第二章主要讨论了如下混合边界条件下的非线性二阶常微分系统 正解存在的几个充分必要条件，并证明所存在的正解具有唯一性，且计算出运用单调迭代方法所得到解的收敛率，最后证明了在上、下解之间存在一个最大解和一个最小解。所用的主要工具是Schauder不动点定理，最大值原理以及单调迭代方法。 第三章研究如下三点边界条件下的非线性二阶常微分系统 在非线性项的适当假设下，利用上、下解方法，给出了此问题正解存在的两个充分必要条件。研究当中最主要的困难在于构造出合适的上、下解，并且所用的方法也可以讨论其它微分系统正解的存在性。 Other Abstract In this paper, using the method of upper and lower solution in the nonlinear functional analysis and matching with monotone iteration, we study the necessary and sufficient condition for existence of the positive solution of two classes nonlinear ordinary differential system. The first chapter represents the problem’s background and the evolution. In the second chapter, the following nonlinear seconder-order ordinary differential system with mixed boundary value condition is studied By constructing appropriate upper and lower solutions sequence, we obtained the necessary and sufficient condition of the existence of the positive solution and proved the uniqueness of the solution; the convergence rate of the solution obtained by the monotone iteration method are computed. Furthermore, we proved that there exist a maximal solution and a minimal solution between the upper solution and the lower solution which we constructed. The main tools are Schauder fixed point theorem, maximum principle and monotone iteration method. In the third chapter, the following second nonlinear second-order ordinary differential system is studied, Under appropriate hypothesis of the nonlinear term, by the upper and lower solution method, we also obtained two necessary and sufficient condition of the existence of positive solution. The difficulty is to construct the appropriate upper and lower solution, the method we used can also discuss the existence of solution for the other differential system. URL 查看原文 Language 中文 Document Type 学位论文 Identifier https://ir.lzu.edu.cn/handle/262010/225454 Collection 数学与统计学院 Recommended CitationGB/T 7714 唐录天. 非线性二阶常微分系统边值问题正解存在的充分必要条件[D]. 兰州. 兰州大学,2008.
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